Clearly, the circle \({\mathbb {S}}^1\)  plays a special role in many constructions,
as the rich theory for central extensions of loop groups and the
Virasoro group shows. Beyond \({\mathbb {S}}^1\) , it seems that
Lie algebras of conformal vector fields
(as generalizations of \(\mathcal {V}({\mathbb {S}}^1)\) ) (cf. Example REF , [1]} and [2]}) and Lie algebras of
sections of vector bundles over Lorentzian manifolds
(or more general “causal” spaces, [3]})
are natural candidates to be investigated
with respect to the existence of semibounded representations.
The latter class of Lie algebras is a natural
generalization of loop algebras.
Here an interesting point is that, although the conformal
groups \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^1) = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\)  of the circle and the conformal
group \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^{1,1})\)  of the Lorentzian torus
are infinite dimensional ,
for Lorentzian manifolds of dimension \(\ge 3\)  the conformal groups are finite
dimensional (cf. [4]}). In particular, it is contained in the list
of [1]}. This leads to the well-studied class of hermitian
Lie groups (see in particular [2]}, [7]}, [3]}).
